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Question #43674

Difficulty: 8/10 | 🟢 Free

The Question

A fully stripped heavy ion of atomic number $Z$ and mass number $A$ moves with a constant highly relativistic velocity $v$ (Lorentz factor $\gamma = (1 - v^2/c^2)^{-1/2} \gg 1$ ) along the $x$ -axis. It passes by a stationary proton (charge $e$ , mass $m_p$ ) located at the origin. The impact parameter is such that the surface of the ion just grazes the proton at the point of closest approach. The rest-frame radius of the ion is given by $R = R_0 A^{1/3}$ . Assuming the proton's displacement during the interaction is negligible, what is the magnitude of the total momentum transferred to the proton?

Options

A.
$\frac{\gamma Z e^2}{2 \pi \epsilon_0 v R_0 A^{1/3}}$
B.
$\frac{Z e^2}{2 \pi \epsilon_0 v R_0 A^{1/3}}$
C.
$\frac{Z e^2}{2 \pi \epsilon_0 \gamma v R_0 A^{1/3}}$
D.
$\frac{\gamma^2 Z e^2}{2 \pi \epsilon_0 v R_0 A^{1/3}}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Nuclear Radius Scaling|Principle of Superposition (Electric Charges)|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the understanding of relativistic electromagnetism and kinematics.

The primary trap is twofold: (1) Students incorrectly apply Lorentz length contraction to the transverse radius of the ion, assuming the impact parameter becomes $R/\gamma$ . (2) Students assume that because the peak transverse electric field is boosted by a factor of $\gamma$ ( $E_{\perp, max} \propto \gamma$ ), the total momentum transfer must also scale with $\gamma$ .

They fail to recognize that the effective interaction time is reduced by $1/\gamma$ due to the longitudinal 'pancaking' of the electric field lines.

The elegant solution uses Gauss's Law to show that the integral of the transverse field over distance (and thus time) is invariant, making the total momentum transfer independent of the Lorentz factor.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Nuclear Radius Scaling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free